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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.18365 |
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| _version_ | 1866918457106235392 |
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| author | Yao, Zhufeng |
| author_facet | Yao, Zhufeng |
| contents | Let $Γ\subset \mathsf{PGL}(d,\mathbb{R})$ be an irreducible projective Anosov subgroup and let $Λ^1(Γ)$ be its projective limit set. Viewing $Λ^1(Γ)$ as an analogue of a self-affine set, we investigate the Hausdorff dimension of $Λ^1(Γ)$ under specific assumptions regarding its affine complexity:
1. If $Λ^1(Γ)$ is of full Hausdorff dimension, then $d= 2$ and $Γ$ is a cocompact lattice.
2. If $d = 3$ and $Γ$ is the image of a closed surface group under an irreducible Anosov representation, then $Λ^1(Γ)$ never has Hausdorff dimension $1$ unless the representation is Hitchin.
3. If the limit set $Λ^1(Γ)$ exhibits a partial quasi-self-similarity (in the sense of Falconer~\cite{falconerselfsimilar1}) -- which can be implied by the ``regular distortion property'' of $Γ$ -- then the Hausdorff dimension of $Λ^1(Γ)$ equals the critical exponent of the first simple root. An application of this result is the computation of the Hausdorff dimension of the limit set for arbitrary $Θ$-positive representations of convex cocompact Fuchsian groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18365 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hausdorff Dimension of Anosov Subgroups' Limit Sets with Special Self-Affine Complexity Yao, Zhufeng Differential Geometry 22E40 Let $Γ\subset \mathsf{PGL}(d,\mathbb{R})$ be an irreducible projective Anosov subgroup and let $Λ^1(Γ)$ be its projective limit set. Viewing $Λ^1(Γ)$ as an analogue of a self-affine set, we investigate the Hausdorff dimension of $Λ^1(Γ)$ under specific assumptions regarding its affine complexity: 1. If $Λ^1(Γ)$ is of full Hausdorff dimension, then $d= 2$ and $Γ$ is a cocompact lattice. 2. If $d = 3$ and $Γ$ is the image of a closed surface group under an irreducible Anosov representation, then $Λ^1(Γ)$ never has Hausdorff dimension $1$ unless the representation is Hitchin. 3. If the limit set $Λ^1(Γ)$ exhibits a partial quasi-self-similarity (in the sense of Falconer~\cite{falconerselfsimilar1}) -- which can be implied by the ``regular distortion property'' of $Γ$ -- then the Hausdorff dimension of $Λ^1(Γ)$ equals the critical exponent of the first simple root. An application of this result is the computation of the Hausdorff dimension of the limit set for arbitrary $Θ$-positive representations of convex cocompact Fuchsian groups. |
| title | Hausdorff Dimension of Anosov Subgroups' Limit Sets with Special Self-Affine Complexity |
| topic | Differential Geometry 22E40 |
| url | https://arxiv.org/abs/2604.18365 |